Systems and methods for optical transmission using supermodes

ABSTRACT

In some embodiments, coupled multi-core fiber is used for optical transmission. The coupled multi-core fiber includes multiple cores each supporting a spatial mode. The cores are positioned close enough to cause coupling between their modes that generates supermodes, that are used to transmit data.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims priority to U.S. Provisional Application Ser.No. 61/522,970, filed Aug. 12, 2011, which is hereby incorporated byreference herein in its entirety.

BACKGROUND

Capacity limits for single-mode fiber transmission has been a subject ofresearch ever since it was recognized that the Kerr nonlinearity imposesa fundamental limit on fiber capacity. It is well known that thenonlinear coefficient is inversely proportional to the effective area ina single-mode fiber. Therefore, a simple and effective way to reduce thenonlinear penalty is to increase the fiber core diameter and thusenlarge the effective area. However, this approach is limited byincreased macro-bending loss and/or dispersion.

Recently, a new method using “few-mode fibers” in single-mode operationwas proposed to increase the core diameter without changing the loss anddispersion properties. Few-mode fibers (FMF) are optical fibers thatsupport more than one spatial mode but fewer spatial modes thanconventional multi-mode fibers. Although FMFs can carry more than onemode, the fundamental mode can be excited and transmitted without modecoupling over very long distances, as long as the effective indexes ofthe supported modes are sufficiently different from each other.

Space-division multiplexing (SDM) has also been proposed to increasefiber capacity. Fiber bundles are attractive for use in SDM because oftheir simplicity and compatibility. FMF is also a candidate for use inSDM because it supports a few large effective area modes and becausemode coupling can be avoided if a large effective index difference(ΔN_(eff)) exists among the modes. Unfortunately, there are somedrawbacks associated with using FMFs for long-distance SDM. First, thereis typically a large differential modal group delay (DMGD) among themodes of FMFs. Second, the modal loss of FMFs increases with mode order.Third, mode coupling is inevitable when using FMFs as the number ofmodes increases because large effective index differences are difficultto maintain for all modes.

Multi-core fiber (MCF) has further been proposed as a candidate for SDMdue to its zero DMGD, equal loss, and ultra-low crosstalk between modes.However, the mode density of MCFs typically must be kept quite low inorder to maintain low crosstalk. For example, the first MCF demonstratedfor SDM transmission was painstakingly fabricated to reduce thecrosstalk to a level of less than −30 dB/km. In more recent efforts,crosstalk has been reduced to the current record of −90 dB/km. Inaddition, each mode of an MCF still suffers a large nonlinear penaltybecause its effective area is the same as or smaller than that of anSMF.

As can be appreciated from the above discussion, it would be desirableto have a high-capacity optical transmission approach that avoids one ormore of the above-described drawbacks.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee. The present disclosure may be better understoodwith reference to the following figures. Matching reference numeralsdesignate corresponding parts throughout the figures, which are notnecessarily drawn to scale.

FIG. 1 is a schematic cross-sectional view of an embodiment of a coupledmulti-core fiber.

FIGS. 2A-2D are images of the field distributions of first, second,third, and fourth supermodes, respectively, for an example four-corecoupled multi-core fiber.

FIG. 3A is a graph of effective area A_(eff) index difference ΔN_(eff)versus pitch-to-core ratio (d/r) for the fundamental mode of six-corecoupled multi-core fiber.

FIG. 3B. is an image of the field distribution of the fundamental modeof the six-core coupled multi-core fiber of FIG. 3A.

FIG. 3C is a graph of A_(eff) versus ΔN_(eff) for a six-core coupledmulti-core fiber and a six-mode few-mode fiber.

FIGS. 4A-4F are images of the field distributions of the first, second,third, fourth, fifth, and sixth supermodes for another example six-corecoupled multi-core fiber.

FIG. 5A is a graph of A_(eff) versus ΔN_(eff) for a six-core coupledmulti-core fibers and a six-mode few-mode fiber.

FIG. 5B is a graph of confinement factor versus ΔN_(eff) for thesix-core coupled multi-core fiber and six-mode few-mode fiber.

FIGS. 6A-6C are graphs of

$\frac{\mathbb{d}c}{\mathbb{d}\omega}$with normalized frequency V=1.6, 1.7, and 1.9, respectively.

FIG. 6D is a graph of

$\frac{\mathbb{d}\;}{\mathbb{d}\lambda}\left( \frac{\mathbb{d}c}{\mathbb{d}\omega} \right)$with V=1.

FIG. 7A is a graph of maximum differential modal group delay (DMGD)versus wavelength at V=1.707 at 1.55 μm and ΔN_(eff)=0.06%.

FIGS. 7B-7D are images of the field distribution of the first, second,and third supermodes, respectively, of a three-core coupled multi-corefiber.

FIGS. 8A-8I are schematic cross-sectional views of multiple alternativeembodiments of a coupled multi-core fiber.

FIG. 9 is a block diagram of a first optical transmission system thatincorporates coupled multi-mode fiber.

FIG. 10 is a block diagram of a second optical transmission system thatincorporates coupled multi-mode fiber.

DETAILED DESCRIPTION

As described above, fiber bundles, multi-core fibers (MCFs), andfew-mode fibers (FMFs) suffer from drawbacks that make them less thanideal for high-capacity optical transmission. Disclosed herein, however,is an approach that avoids many of these drawbacks. That approachexploits mode coupling between the cores of a “coupled multi-core fiber”(CMCF) to generate supermodes that extend beyond the boundaries of theindividual cores. Unlike conventional multi-core fiber, the core-to-coredistance of CMCF is small to encourage such mode coupling. The closerspacing of the cores increases the mode density because the modes aresupported by a fiber having a smaller cross-section. In addition, thecloser spacing leads to a larger mode effective area than that ofconventional MCF and FMF. Simulations have shown that the disclosed CMCFprovides lower modal dependent loss, mode coupling, and differentialmodal group delay than FMFs. These results suggest that CMCF is a goodcandidate for both single-mode operation (SMO) and space-divisionmultiplexing (SDM) operation.

In the following disclosure, various embodiments are described. It is tobe understood that those embodiments are example implementations of thedisclosed inventions and that alternative embodiments are possible. Allsuch embodiments are intended to fall within the scope of thisdisclosure.

Theory

The basic supermode analysis of CMCFs will now be discussed. A four-coreCMCF, shown in FIG. 1, is selected as an example. The cores of the CMCFare assumed to be identical and each of them supports only one mode. Theradius of the cores is r, and the distances between adjacent cores andnon-adjacent cores are d₁ and d₂, respectively. The cores and thecladding that surrounds them have refractive indices of n₁ and n₂,respectively. The mode of each core has the same normalized frequency(V-number)

$V = {\frac{2\pi}{\lambda_{0}}r{\sqrt{n_{1}^{2} - n_{2}^{2}}.}}$

According to coupled-mode analysis, the interaction between the modes ofthe four individual cores can be described by the following coupled-modeequation

$\begin{matrix}{{{\frac{\mathbb{d}\;}{\mathbb{d}z}A} = {{- j}\;\overset{\_}{M}\; A}},{{\frac{\mathbb{d}\;}{\mathbb{d}z}A} = {{- j}\;\overset{\_}{M}\; A}}} & (1)\end{matrix}$where A=[A₁ A₂ A₃ A₄]^(T),

${\overset{\_}{M} = \begin{pmatrix}\beta_{0} & c_{1} & c_{2} & c_{1} \\c_{1} & \beta_{0} & c_{1} & c_{2} \\c_{2} & c_{1} & \beta_{0} & c_{1} \\c_{1} & c_{2} & c_{1} & \beta_{0}\end{pmatrix}},$A_(i) (i=1, 2, 3, 4) refers to the complex amplitude of the electricalfield of the ith core, β₀ is the propagation constant of the singlemode, and c₁ and c₂ are the coupling coefficients between adjacent andnon-adjacent cores, respectively. Even though it has, for simplicity,been assumed that the cores have the same propagation constant, thephysics described here also apply to cores with slightly differentpropagation constants. Coupling length as well as the couplingcoefficient are frequently used to more directly describe the amount ofcoupling. Coupling length is defined as π/2c, where c is couplingcoefficient. Since M is Hermitian for a lossless system, it can bediagonalized by a unitary matrix such thatQ ⁻¹ MQ=Λ, Q ⁻¹ MQ=Λ  (2)where Λ is a diagonal matrix,

$\begin{matrix}{\Lambda = \begin{pmatrix}\beta_{1} & 0 & 0 & 0 \\0 & \beta_{2} & 0 & 0 \\0 & 0 & \beta_{3} & 0 \\0 & 0 & 0 & \beta_{4}\end{pmatrix}} & (3)\end{matrix}$in which β_(m) (m=1, 2, 3, 4) is the propagation constant of the mthsupermode supported by the CMCF. The amplitude matrix for the supermodesis represented asB=Q⁻¹A  (4)under which the coupled-mode Eq. (1) reduces to

$\begin{matrix}{{\frac{\mathbb{d}\;}{\mathbb{d}z}B} = {{{- j}\;\Lambda\;{B.\mspace{14mu}\frac{\mathbb{d}\;}{\mathbb{d}z}}B} = {{- j}\;\Lambda\; B}}} & (5)\end{matrix}$Under the weakly guiding approximation, a general expression of thecoupling coefficient c_(j) (j=1, 2) is given as

$\begin{matrix}{c_{j} = {\sqrt{\frac{n_{1}^{2} - n_{2}^{2}}{n_{1}^{2}}} \cdot \frac{1}{r} \cdot \frac{U^{2}}{V^{3}} \cdot \frac{K_{0}\left( {W\;{d_{j}/r}} \right)}{K_{1}^{2}(W)}}} & (6)\end{matrix}$where U and W can be found by solving equation U·K₀(W)·J₁(U)=W·K₁(W)·J₀(U) and U²+W²=V². The J's and the K's are Bessel functions of the firstkind and modified Bessel functions of the second kind, respectively.After obtaining the coupling coefficients c₁ and c₂, the supermodes canbe solved as eigen-modes. The propagation constant of the supermodes arethe eigenvalues, given by:β₁=β₀+2c ₁ +c ₂;β₂=β₀ −c ₂;β₃=β₀ −c ₂;β₄=β₀−2c ₁ +c ₂.  (7)

When the coupling coefficients are small, the supermodes have similarpropagation constants. These supermodes will couple to each other afterpropagation. SDM, in the form of mode-division multiplexing (MDM), usingthese supermodes has the advantage that the modal dispersion is smalland therefore mode crosstalk can be efficiently computationallydecoupled using multiple-input multiple-output (MIMO) equalizationalgorithms. A fiber that achieves a small coupling constant can bedesigned using Eq. (6). For the particular case of the four-core CMCF,the second and third supermodes are degenerate and have the samepropagation constants. Mode-division multiplexing using these degeneratesupermodes also has the advantage that mode crosstalk can becomputationally efficiently de-coupled using multiple-inputmultiple-output equalization algorithms. Using n₁=1.47, n₂=1.468, r=7μm, d=14 μm and λ=1.55 μm, the field distributions of each supermode canbe calculated. Those field distributions are shown in FIGS. 2A-2D.

One important characteristic of supermodes in four-core CMCFs is thatthey are superpositions of isolated modes with equal amplitude but notalways the same phase. The fundamental supermode is shown in FIG. 2A andis the in-phase mode with the largest propagation constant. Thehigher-order supermodes have the field reversals between adjacent ornon-adjacent core regions, as is shown in FIGS. 2B-D. Thisequal-amplitude characteristic gives similar confinement factors for thesupermodes, leading to very small modal dependent loss. It is clear fromthis example that the properties of supermodes are determined not onlyby the parameters of each core but the pitch (i.e., distance) betweencores. In other words, CMCFs have more degrees of freedom or largedesign space than MCFs and FMFs.

Fiber Design and Application

As identified above, CMCFs can be designed and applied to both SMO andMDM applications. For each case described below, a specific design ofCMCF will be presented and compared to FMF in terms of transmissionperformance.

In order to achieve better performance, i.e., reduced nonlinear penalty,in the single-mode operation, the fundamental supermode should have alarge effective area A_(eff) given by

$\begin{matrix}{A_{eff} = \frac{{{\int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{{I\left( {x,y} \right)}{\mathbb{d}x}{\mathbb{d}y}}}}}^{2}}{\int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{{I^{2}\left( {x,y} \right)}{\mathbb{d}x}{\mathbb{d}y}}}}} & (8)\end{matrix}$where I(x, y) is the mode intensity distribution. To reduce or minimizesupermode coupling and guarantee single-mode operation, the fundamentalsupermode should also have a large difference in effective indexΔN_(eff), which is given by

$\begin{matrix}{{\Delta\;{N_{eff}\left( {i,j} \right)}} = {{\frac{1}{k_{0}} \cdot \left( {\beta_{i} - \beta_{j}} \right)} = {\frac{1}{k_{0}} \cdot {\sum\limits_{n}^{\;}{\left( {a_{n}^{i} - a_{n}^{j}} \right) \cdot c_{n}}}}}} & (9)\end{matrix}$where i and j represent the supermode number, a_(n) ^(i) and a_(n) ^(j)denote the coefficient of the coupling coefficient c_(n) in theexpression of β for the ith and jth supermode respectively (see Eq. (7)as an example).

The number of modes and the bending loss of the large-area fundamentalsupermode are two of the most important fiber properties. For an examplecomparison between the CMCFs and FMFs, the macro-bending loss has beenselected to be 0.0308 dB/m at a mandrel radius of 20 mm and the numberof modes has been selected to be six for both types of fibers. Thebending loss value is calculated using the curvature loss formula andits value is set in accordance with that for standard SMF fibers. Notethat this value represents the minimum bending loss as other factorsincluding micro-bending loss and the situation in which manufactureimperfections are not considered. Both the six-core CMCF and thesix-mode FMF have two pairs of degenerate modes and two non-degeneratemodes. The cores of the six-core CMCF are arranged to optimize A_(eff)of the fundamental supermode. The fundamental supermode fielddistribution as well as the core arrangement is shown in FIG. 3B. Forthe purposes of this discussion, all designs are based on step-indexprofiles. As a result, the design variables include the index differenceΔN_(eff) and the core radius r for both CMCFs and FMFs, and an extraparameter, namely, the pitch-to-core ratio (d/r) for CMCFs in additionto the core arrangement.

Before presenting the detailed comparison, it is worthwhile to considerthe relationship between the pitch-to-core ratio (d/r) and ΔN_(eff) inCMCFs. From Eq. (6) and (9), it can be appreciated that a reduced d/rvalue would increase the coupling coefficient c, and result in a largeΔN_(eff). FIG. 3A shows the dependence of A_(eff) and ΔN_(eff) on d/r.As cores are moved closer, increased coupling between the cores inducesa larger split of the effective indexes of supermodes. It is observedthat A_(eff) has a weak dependence on d/r while ΔN_(eff) changes sharplywith d/r. Therefore the smallest value of d/r (d/r=2) is chosen for thiscomparison.

The fundamental mode properties for the six-core CMCFs and six-mode FMFswith the same macro-bending loss are identified in FIG. 3C. That figurereveals that CMCFs perform better in terms of both A_(eff) and ΔN_(eff).An important reason for the larger A_(eff) with CMCFs is that thefundamental supermode is the in-phase superpostition of the modes of sixisolated cores. The larger ΔN_(eff) is mainly attributed to theoptimization of the extra design freedom d/r. To provide a morecomprehensive and detailed comparison, ΔN_(eff) is set at a sufficientlylarge value 1e-3 for both fibers while other important fiber parametersare listed in Table 1. The A_(eff) of the CMCF is increased by 60%without compromising other properties. It is noted that slight differentindex differences (ΔN_(eff) s) are applied to CMCFs and FMFs in order tomaintain the same number of modes for both fibers.

TABLE 1 Properties of coupled multi-core fiber and few-mode fiber designfor single-mode operation Step-index Step-index @1.55 μm CMCF FMF Modenumber 6 6 Index 0.34% 0.23% difference Core area 357 357 (μm²)Confinement 95.8% 95.2% factor ΔN_(eff) 1e−3 1e−3 A_(eff) (μm²) 438 274Bending loss 0.0308 0.0308 20 mm φ (dB/m) Dispersion 19.2 22.7(ps/nm/km) Dispersion 0.077 0.064 slope (ps/nm²/km)

It is expected that SDM optical transmission can operate successfullyeither without supermode coupling or with supermode coupling but withnegligible or small differential modal group delay (DMGD). For the casein which there is no mode coupling, modes propagate independently andtherefore can be separately detected. For the case in which there ismode coupling but small DMGD, modes may couple to each other but theycan be detected together and then separated by using MIMO based digitalsignal processing (DSP) techniques. These two methods can be combined insupermode multiplexing as will be explained below.

In the next simulation, the number of modes is again selected to be six.However, the core arrangement is without a center core so that thehigher order supermodes and the fundamental supermode are moresymmetrical. The field distributions of the supermodes are shown in FIG.4. Again, both six-core CMCFs and six-mode FMFs support six modesincluding two pair of degenerate modes and two other non-degeneratemodes. For the six-mode FMFs, the two pairs of degenerate modes are thedegenerate LP₁₁ and LP₂₁ modes. For the CMCFs, the two pairs ofdegenerate modes are the second and third, fourth, and fifth supermodes.The degenerate supermodes have identical effective indexes and thusthere is no DMGD between them. The non-degenerate supermodes havedifferent effective indexes. Fortunately, these non-degeneratesupermodes/supermode groups can be designed to have low crosstalk bymaintaining a large ΔN_(eff) between them. Therefore demultiplexing inSDM using CMCFs can be successfully performed in two steps: (i) thenon-degenerate supermodes/supermode groups are separately detected whilethe degenerate supermodes are still mixed together, and (ii) mixedsignals in the degenerate supermodes are recovered by the MIMO-based DSPtechniques.

There are three design parameters that can be used to optimize theperformance for SDM: (1) ΔN_(eff) between any two modes should besufficiently large to avoid mode coupling, (2) mode losses should besimilar to each other and as low as possible, and (3) large effectiveareas are always required for reducing nonlinearity. Based on theseparameters, six-core CMCFs and six-mode FMFs are designed respectivelyand their performances are shown in FIGS. 5A and B. The macro-bendinglosses of the fundamental modes for both fibers are fixed at 0.0308 dB/mat a mandrel radius of 20 mm. A confinement factor is used here tocharacterize the mode loss. A higher confinement factor implies lowerloss as it indicates less bending loss. From FIGS. 5A and B, it can beappreciated that CMCFs show a significant advantage of attaining largeΔN_(eff), confinement, and A_(eff) for all supermodes. In other words,the supermodes tend to preserve less mode coupling, lower loss, andlower nonlinearity than regular modes. All supermodes have similarproperties (including mode coupling, loss, and nonlinearity), which isimportant for long-distance mode-division multiplexing. Higher-ordermodes in FMF appear to have larger effective areas in FIG. 5A, but theselarge effective areas actually result from low confinement (as indicatedin FIG. 5B) and hence have no practical benefit.

A CMCF design with zero or small DMGD between supermodes has also beenevaluated. In this case, even though supermode coupling may still exist,signal travel along the supermodes is at the same/similar groupvelocities. Therefore, the supermodes could be detected together anddemultiplexing can be performed using MIMO DSP techniques as mentionedabove. According to Eq. (6), DMGD between the ith and jth supermodes,can be represented as

$\begin{matrix}{{D\; M\; G\;{D\left( {i,j} \right)}} = {{\frac{\mathbb{d}\beta_{i}}{\mathbb{d}\omega} - \frac{\mathbb{d}\beta_{j}}{\mathbb{d}\omega}} = {\sum\limits_{n = 1}^{2}{\left( {a_{n}^{i} - a_{n}^{j}} \right) \cdot \frac{\mathbb{d}c_{n}}{\mathbb{d}\omega}}}}} & (10)\end{matrix}$where a_(n) ^(i) and a_(n) ^(j) the supermode propagation constant β forthe ith and jth supermode, respectively, to the coupling coefficientsc_(n) as given in Eq. (7). Using Eq. (6),

$\frac{\mathbb{d}c_{n}}{\mathbb{d}\omega}\left( {{n = 1},2} \right)$is obtained as,

$\begin{matrix}{\frac{\mathbb{d}c_{n}}{\mathbb{d}\omega} = {\frac{1}{r} \cdot \left\{ {{\frac{\partial\;}{\partial\omega}{\sqrt{1 - \frac{n_{2}^{2}(\omega)}{n_{1}^{2}(\omega)}} \cdot \left\lbrack {\frac{U^{2}}{V^{3}} \cdot \frac{K_{0}\left( {W\;{d_{n}/r}} \right)}{K_{1}^{2}(W)}} \right\rbrack}} + {\sqrt{1 - \frac{n_{2}^{2}}{n_{1}^{2}}} \cdot {\frac{\partial\;}{\partial\omega}\left\lbrack {\frac{{U(\omega)}^{2}}{{V(\omega)}^{3}} \cdot \frac{K_{0}\left( {{W(\omega)} \cdot {d_{n}/r}} \right)}{K_{1}^{2}\left( {W(\omega)} \right)}} \right\rbrack}}} \right\}}} & (11)\end{matrix}$

It should be noted that Eq. (10) is presented for four-core CMCFs, inwhich DMGD is a linear combination of

$\frac{\mathbb{d}c_{1}}{\mathbb{d}\omega}\mspace{14mu}{and}\mspace{14mu}\frac{\mathbb{d}c_{2}}{\mathbb{d}\omega}$with different weighting coefficients for different supermodes. It isclear that, in order to achieve zero DMGD among all the supermodes, both

$\frac{\mathbb{d}c_{1}}{\mathbb{d}\omega}\mspace{14mu}{and}\mspace{14mu}\frac{\mathbb{d}c_{2}}{\mathbb{d}\omega}$should vanish, which is unlikely if not impossible to realize in asimple step-index profile CMCF. This problem also exists for other CMCFstructures where the number of cores is more than three. Therefore,three-core CMCFs are chosen here for zero DMGD design as they onlycontain adjacent core coupling, i.e., only one value of c exists. As aresult, total DMGD scales with

$\frac{\mathbb{d}c_{1}}{\mathbb{d}\omega}$and it is equivalent to attain zero for

$\frac{\mathbb{d}c_{1}}{\mathbb{d}\omega}$in order to achieve zero DMGD in this structure. The mode fields ofthree-core CMCFs are given in FIGS. 7B, C, and D. As shown in Eq. (11),

$\frac{\mathbb{d}c}{\mathbb{d}\omega}$comprises two parts: a frequency dependent index (n₁, n₂) component anda frequency dependent waveguide parameters (V, U, W) component, i.e.,the material and waveguide DMGD. At first glance, one might believe thatmaterial DMGD is larger than waveguide DMGD (material dispersion isdominant in chromatic dispersion of standard SMFs). However, it isincorrect to draw an analogy between DMGD and chromatic dispersionbecause the nature of DMGD is differential group delay (DGD) betweenmodes instead of dispersion within one mode. The fact that allsupermodes propagate in the same material but with different propagationconstants implies that the material DMGD should be negligible comparedto waveguide DMGD. This conclusion has been verified by simulation.Since material DMGD is significantly smaller than waveguide DMGD, theywill be neglected in the following discussion to simplify the analysis.

As indicated in Eq. (11),

$\frac{\mathbb{d}c}{\mathbb{d}\omega}$is determined by the pitch-to-core ratio (d/r), V-number, and coreradius r (or equivalently, V-number and index difference ΔN_(eff) since

$\left. {V = {\frac{2\pi}{\lambda_{0}}r\sqrt{n_{1}^{2} - n_{2}^{2}}}} \right).$The relationship between those variables is shown in FIGS. 6A, B, and C.It is confirmed by both analysis and simulation that when the V-numberis fixed, zero DMGD is attained if and only if d/r reaches a certainvalue. As the V-number increases, zero DMGD is realized for a smallervalue of d/r, as indicated in FIGS. 6A, B, and C. Therefore, in order toattain zero DMGD, the V-number is limited to below 1.71 because d/rcannot less than 2. This is demonstrated by the zero DMGD horizontallines and their locations with different V-numbers in FIGS. 6A, B, and C(in FIG. 6C, with V-number>1.71, the zero DMGD line does not exist).Apart from zero DMGD, a sufficiently small DMGD is enough for practicaluse as well. This can be obtained by reducing index difference ΔN_(eff).

To meet the practical application requirements in a wavelength-divisionmultiplexing (WDM) system, CMCFs further require small DMGD variationwithin a certain range of wavelength, i.e., a small modal differentialgroup delay slope (DMGDS). DMGDS can be considered to be linear within anarrow range of wavelength. Similar to DMGD, DMGDS between the ith andjth supermodes in a three-core structure can be represented as

$\begin{matrix}{{D\; M\; G\; D\;{S\left( {i,j} \right)}\frac{\mathbb{d}\;}{\mathbb{d}\lambda}\left( {\frac{\mathbb{d}\beta_{i}}{\mathbb{d}\omega} - \frac{\mathbb{d}\beta_{j}}{\mathbb{d}\omega}} \right)} = {{\left( {a_{1i} - a_{1j}} \right) \cdot \frac{\mathbb{d}\;}{\mathbb{d}\lambda}}\left( \frac{\mathbb{d}c_{1}}{\mathbb{d}\omega} \right)}} & (12)\end{matrix}$Given that material DMGD is negligible, the

$\frac{\mathbb{d}\;}{\mathbb{d}\lambda}\left( \frac{\mathbb{d}c}{\mathbb{d}\omega} \right)$term can be further expressed as

$\begin{matrix}{{\frac{\mathbb{d}\;}{\mathbb{d}\lambda}\left( \frac{\mathbb{d}C}{\mathbb{d}\omega} \right)} = {\frac{1}{r} \cdot \sqrt{1 - \frac{n_{2}^{2}}{n_{1}^{2}}} \cdot {\frac{\partial^{2}}{{\partial\lambda} \cdot {\partial\omega}}\left\lbrack {\frac{{U(\omega)}^{2}}{{V(\omega)}^{3}} \cdot \frac{K_{0}\left( {{W(\omega)} \cdot {d/r}} \right)}{K_{1}^{2}\left( {W(\omega)} \right)}} \right\rbrack}}} & (13)\end{matrix}$

DMGDS is plotted vs. d/r and ΔN_(eff) in FIG. 6( d). Even though zeroDMGDS can be realized, they occur at a larger value of d/r with respectto zero DMGD. Therefore, it is difficult to achieve zero DMGD and DMGDSsimultaneously. Even so, DMGDS can still be reduced by decreasing indexdifference ΔN_(eff). Wavelength-dependent DMGD as well as mode fields ofa specific three-core CMCF design is provided in FIGS. 7A and B. TheDMGD is below 60 ps/km over the entire C band, which is the same valueachieved by three-mode fiber using a depressed cladding index profile.

Discussion

The possible candidates of next generation transmission fibers,especially for the application of SDM, will next be discussed. Thesefibers include MCF, FMF, fiber bundle and the disclosed CMCF. It isnoted that, even though fiber bundle is usually referred as a bundle ofSMFs, a bundle could be a bundle of MCFs, FMFs, or CMCFs. In otherwords, the fiber bundle is only a concept of package form, not a type offiber. Therefore, only MCF, FMF, and CMCF are compared with each otheras shown in Table 2.

TABLE 2 Comparison of next generation transmission fibers for spatialdivision multiplexing Multi- core Coupled Few-mode fiber multi-corefiber (MCF) fiber (CMCF) (FMF) Transmission Spatial Mode Low High Highproperty Density^(a) Differential Zero Small Large Modal Group(controllable) Delay (DMGD) Crosstalk^(b) Low Easy to Hard to ControlControl Modal Dependent Equal Similar Different Loss Loss Low Low High(higher order modes) Effective Area Small Large Large (A_(eff))^(c)Amplifi- Pump coupling Hard Easy Easy cation Power efficiency Low HighHigh (cladding pump) Scalability Good Good Bad Inter-connect Hard HardEasy ^(a)Number of spatial modes per unit area of the fibercross-section. ^(b)Crosstalk between spatial modes (single modes forMCF; supermodes for CMCF; regular modes for FMF). ^(c)Effective area ofspatial modes.

The transmission property is a critical property because it determinesthe fundamental capacity of the fiber. For SDM transmission, thecapacity of the system scales with the number of modes and hence isproportional to the spatial mode density. As identified above, thespatial mode density of MCFs is much lower than CMCFs and FMFs becauseof the low crosstalk requirement. Linear crosstalk caused by modecoupling is one of the most critical impairments in both CMCF and FMFsystems. In order to facilitate spatial demultiplexing and linearcrosstalk cancelation, increasing ΔN_(eff) to reduce mode coupling orequalizing DMGD to lessen computation load of the MIMO process should beconsidered in fiber design. As has been established above, CMCFs havemore degrees of design freedom, namely, pitch length and corearrangement, and thus improve both ΔN_(eff) and DMGD significantly. Lowand equal modal loss is another advantage of CMCFs over FMFs. Similar toMCFs, light is well confined within each core of CMCFs. The confinementfactors of all supermodes are relatively high and similar. In contrast,high-order modes in FMFs have much larger bending loss than thefundamental mode indicated by low confinement factors. In addition,CMCFs can have larger A_(eff) spatial mode than MCFs. Therefore,nonlinear impact is directly reduced which leads to higher fundamentalcapacity of the systems.

For long-haul SDM transmission, a low noise and power efficientamplifier is desirable. The design of the fiber amplifier should bematched with the type of transmission fiber. In the second part of Table2, the complexity and performance of amplification are compared amongthe three candidate fibers. To reduce amplified stimulated emissionnoise, most doped ions must be inverted, which requires a pump powerscaled with the effective guiding area of the pump. For the case of acladding pump, a strong pump is guided in the inner cladding of theactive fiber. With closer spacing between cores, a CMCF amplifier(CMCFA) is expected to have much smaller inner cladding size as comparedto that of an MCF amplifier (MCFA). Consequently, the operating pumppower can be dramatically reduced. In order to increase power efficiencyfor MCFAs, the pump may need to be launched core-by-core usingfree-space optics. However, the launching scheme requires furtheralignment complexity.

Another unique property of CMCFs and MCFs is that the number of spatialmodes is the same as the core number of the fiber. Any number of spatialmodes can be obtained by adjusting the core number. In contrast, forFMFs, tuning either core size or index difference cannot alwaysguarantee the desired number of modes. Achieving the desired number ofmodes becomes more problematic as the number of modes increases. Forexample, FMFs can never support four modes including degenerate modes.

Example Embodiments

FIGS. 8A-F illustrate multiple embodiments of CMCFs 10. In eachembodiment, the CMCF 10 includes at least two cores 14 that aresurrounded by a cladding 16. More particularly, the embodiment of FIG.8A comprises two cores 14, the embodiment of FIG. 8B comprises threecores 14, the embodiment of FIG. 8C comprises four cores 14, theembodiment of FIG. 8D comprises five cores 14, the embodiment of FIG. 8Ecomprises six cores 14, and the embodiment of FIG. 8F comprises sevencores 14. As can be appreciated from the figures, the cores 14 can bearranged in various layouts. For example, the center of the CMCF 10 caninclude a core 14, as in the embodiment of FIG. 8F, or can omit a core,as in the embodiment FIG. 8E. The former arrangement may be more usefulin single mode operation while the latter may be more useful insupermode multiplexing.

In some embodiments, one or more of the cores 14 is a single mode corethat only supports a single spatial mode. In such a case, the core 14and can have a diameter of approximately 8 to 10 μm. In otherembodiments, one or more of the cores 14 is a multi-mode core, such as afew-mode core, that supports multiple spatial modes. In such a case, thecore 14 can have a diameter of approximately 10 to 50 μm. In otherembodiments (not shown), the cladding can have photonic crystalstructures so that fiber modes are guided by the photonic bandgap of thephotonic crystal cladding. In this case, the core can be hollow orsolid.

As is further shown in each of FIGS. 8A-F, the cores 14 are positionedin close proximity within the cladding 16 so as to encourage coupling oftheir modes. In some embodiments, the cores 14 nearly or actually toucheach other. In that case, the pitch-to-core ratio d/r is approximately2. In other embodiments, the cores 14 can be spaced a greater distancefrom each other. That distance depends upon the degree of mode couplingthat is desired, which is also dependent upon the index differenceΔN_(eff). In general, mode coupling decreases as the core separationincreases, and increases as the index difference decreases. For atypical step-index fiber in which the index difference is approximately0.4%, the pitch-to-core ratio can, for example, be approximately 2 to10.

The degree of mode coupling enabled by the CMCF configuration can bequantified in several ways, including the mode coupling length, the modecoupling coefficient, and the crosstalk that exists between the modes.In some embodiments, the CMCF has a mode coupling length less than 7 km,a coupling coefficient c larger than 0.2 km⁻¹, and a crosstalk largerthan −30 dB/km.

Each core 14 has a refractive index that is greater than that of thecladding 16. It is noted, however, that the index of refraction can varywithin each core 14. For example, the cores 14 can be step index orgradient index cores, if desired. While the cores 14 have the samepropagation constant, they do not need to always have the same indexprofile. In fact, they can have different index profile and radius aslong as c.c>ΔN_(eff).

As shown in FIGS. 8G-I, a CMCF can include more than one cladding. Inthe embodiment of FIG. 8G, a CMCF 20 comprises an inner cladding 22surrounding the cores 24 that has a lower index of refraction than anouter cladding 26. This depressed cladding structure may reduce bendingloss. In the embodiment of FIG. 8H, a CMCF 30 has an inner claddingsurrounding the cores 32 has a higher index of refraction than an outercladding 34, which can further reduce DMGD for the supermodes. It isnoted that the inner cladding can be used to adjust the coupling statebetween the cores without changing the pitch-to-core ratio. This mightbe useful for designing a CMCF 40 that has supermodes formed by core 42that include multiple sub-cores 44, as is shown in FIG. 8I.

Although MCFs with close core spacing have been used in certain laserapplications separate from optical communications, it is noted thatthose MCFs are active fibers that are doped with rare-earth elements toacquire gain. In contrast, the CMCFs described herein, such as those ofFIG. 8, can be passive fibers that are not doped with rare-earthelements and that do not acquire gain.

FIG. 9 illustrates a simple optical transmission system 50 thatincorporates CMCFs. As is shown in that figure, the system 50 comprisesa transmitter 52, a receiver 54, and a length of CMCF fiber 56. Thetransmitter 52 can comprise any transmitter that can transmit opticalsignals along the fiber 56 and the receiver 54 can comprise any receiverthat can receive those transmitted signals. By way of example, thetransmitter 52 comprises a laser with an external modulator and thereceiver 54 at least comprises a photodetector.

FIG. 10 illustrates a further optical transmission system 60, which canbe used as a long-distance communication link. As is shown in FIG. 10,the system 60 also comprises a transmitter 62 and a receiver 64. Inaddition, the system 60 comprises multiple spans 66. One or more of thespans 66 comprises a length of CMCF 68. By way of example, each span 66can be approximately 40 to 50 km in length for undersea applications and80 to 100 km in length for terrestrial applications. The overall lengthof the communication link is potentially infinite but in many cases willextend across thousands of kilometers. Separating each span 66 along thelength of the communication link between the transmitter 62 and thereceiver 64 are optical amplifiers 70 that amplify the optical signalscarried by the CMCFs 68. Although the system 60 has been described andillustrated as comprising an amplifier 70 at the end of each span 66, itis noted that other components, such as repeaters, could be usedinstead. It is noted that multiple communication links such as thatillustrated in FIG. 10 can be connected in such a way to form networks.

Both systems 50 and 60 can be used to optically transmit data, forexample across long distances. In some embodiments, this is performed byencoding data on an optical carrier, exciting a supermode of the CMCFwith the data-encoded optical carrier, optically transmitting thesupermode along the CMCF, and then receiving the transmitted signalusing a receiver. When transmitting using SMO, only a single supermode,such as the fundamental supermode, can be excited. When transmittingdata using MDM, multiple supermodes can be excited. In some embodiments,the number of data streams is the same as the number of supermodes. Inother embodiments, the number of data streams is fewer than the numberof supermodes.

Conclusion

Described herein are coupled multi-core fiber (CMCF) designs forlong-haul transmission. The new designs exploit the coupling between thecores of multi-core fibers instead of avoiding it. The designs haveadvantages over the conventional multi-core fiber in terms of highermode density and larger mode effective area. It is possible to avoidmode coupling between supermodes through additional degree of designfreedom, which includes the pitch-to-core ratio and core arrangement.Although this disclosure has described with particularity four-core,six-core, and three-core structures due to their simplicity andsymmetry, the number and arrangement of the cores can be modified tomeet the requirements of the anticipated end use.

For single-mode operation, CMCFs can attain larger ΔN_(eff) and A_(eff)than FMFs. As a result, CMCFs tend to have less mode coupling andnonlinearity, which is important for efficient long-haul transmission.The excitation of the fundamental supermode, as well as the higher-ordersupermodes, can be realized using free-space optics which have been usedto excite spatial modes of FMFs. There is a possibility that supermodeexcitation is even simpler because supermodes can be seen assuperpositions of modes of the coupled cores. The crosstalk accumulatedin the higher-order supermodes after transmitting a certain distancecould also be removed optically by using the phase reversal of higherorder supermodes. For example, a 4f-configuration component can be usedto remove the power of higher-order supermodes in the Fourier plane. Inthis disclosure, d/r was selected to be the minimum (d/r=2) to optimizeboth ΔN_(eff) and A_(eff). However, it is noted that the value of d/rcan be tuned to meet a specific individual requirement of a largeΔN_(eff) or A_(eff).

For mode-division multiplexing, two concepts can be simultaneouslyapplied in CMCFs. One is to utilize zero DMGD between the two degeneratesupermodes and the other is to eliminate mode coupling between thenon-degenerate supermodes. Demultiplexing can be realized by firstdetecting the non-degenerate supermodes separately and then recoveringsignals mixed in the degenerate supermodes using MIMO DSP techniques.Compared to FMF modes, the supermodes can maintain less mode coupling,nonlinearity, and similar loss. The other possibility is to design CMCFswith zero DMGD between all the supermodes and a DMGD slope small enoughfor WDM system. In this disclosure, a 60 ps/km DMGD between any twosupermodes across the C-band has been demonstrated in a step-indexthree-core CMCF. This DMGD value can be further reduced by reducing theindex difference between the core and cladding. It has been reportedthat graded-index profiles can decrease DMGD in FMFs. It is likely thata more sophisticated index profile including the graded-index profilefor CMCFs may further reduce the DMGD.

The invention claimed is:
 1. A passive, coupled multi-core fibercomprising: multiple cores each supporting a spatial mode, the coresbeing positioned close enough to cause coupling between their modes thatgenerates supermodes having an orthogonal mode distribution, where thesupermodes are capable of transmitting data, the cores excluding amaterial that enables them to acquire gain, where the cores aremulti-mode cores that support multiple spatial modes; and an outercladding surrounding the cores.
 2. The passive, coupled multi-core fiberof claim 1, wherein the fiber has a pitch-to-core ratio of approximately2 to
 10. 3. The passive, coupled multi-core fiber of claim 2, whereineach core and the cladding have an index difference of approximately0.4%.
 4. The passive, coupled multi-core fiber of claim 1, wherein thefiber has a coupling length less than 7 km.
 5. The passive, coupledmulti-core fiber of claim 1, wherein the fiber has a couplingcoefficient larger than 0.2 km⁻¹.
 6. The passive, coupled multi-corefiber of claim 1, wherein the fiber has a crosstalk larger than −30dB/km.
 7. The passive, coupled multi-core fiber of claim 1, wherein themulti-mode cores are few-mode cores that support no more than sevenspatial modes.
 8. The passive, coupled multi-core fiber of claim 1,wherein at least one of the cores has an index of refraction thatradially varies.
 9. The passive, coupled multi-core fiber of claim 1,wherein the fiber comprises an inner cladding that is surrounded by theouter cladding, the inner cladding having an index of refraction that isdifferent from the index of refraction of the outer cladding.
 10. Anoptical transmission system, comprising: one or both of a transmitterand a receiver; and a coupled multi-core fiber including multiple coreseach supporting a spatial mode, the cores being positioned close enoughto cause coupling between their modes that generates supermodes havingan orthogonal mode distribution that are available for transmitting dataover the system, where the cores are multi-mode cores that supportmultiple spatial modes.
 11. The optical transmission system of claim 10,wherein the coupled multi-core fiber has a pitch-to-core ratio ofapproximately 2 to
 10. 12. The optical transmission system of claim 10,wherein each core and cladding have an index difference of approximately0.4%.
 13. The optical transmission system of claim 10, wherein thecoupled multi-core fiber has a coupling length less than 7 km.
 14. Theoptical transmission system of claim 10, wherein the coupled multi-corefiber has a coupling coefficient larger than 0.2 km⁻¹.
 15. The opticaltransmission system of claim 10, wherein the coupled multi-core fiberhas a crosstalk larger than −30 dB/km.
 16. The optical transmissionsystem of claim 10, wherein the multi-mode cores are few-mode cores thatsupport no more than seven spatial modes.
 17. A method of transmittingdata using coupled multi-core fiber, where the cores of the coupledmulti-core fiber are multi-mode cores, the method comprising: encodingdata to be transmitted on an optical carrier; exciting a supermode ofthe coupled multi-core fiber with a data-encoded optical carrier, thesupermode being generated due to mode coupling between the cores of thecoupled multi-core fiber, the supermode having an othrogonal modedistribution; and optically transmitting the supermode along the coupledmulti-core fiber.
 18. The method of claim 17, wherein the coupledmulti-core fiber supports multiple supermodes but only one supermode isexcited so as to transmit data in a single-mode operation scheme. 19.The method of claim 18, wherein the excited supermode is the fundamentalsupermode of the coupled multi-core fiber.
 20. The method of claim 17,wherein the coupled multi-core fiber supports multiple supermodes andmultiple supermodes are excited so as to transmit data in mode-divisionmultiplexing scheme.